The present investigation aims to examine the impact of incorporating rare earth elements (Dy and Nd) into dual-phase hexagonal close-packed (HCP) ternary refractory medium-entropy alloy HfTiZr. The HfTiZr, (HfTiZr)98Dy2, and (HfTiZr)96Dy2Nd2 alloy exhibit yield strength of 356.95, 424.53, and 458.81 MPa, respectively, and ductility of 8.5, 11, and 10% elongation, respectively. A comparative investigation is conducted to analyze the microstructure and mechanical properties of HfTiZr, (HfTiZr)98Dy2, and (HfTiZr)96Dy2Nd2. With the addition of Dy and Nd, the alloy exhibits an increase in BCC phase fraction with basket weave morphology. The micro indentation technique was employed to determine the average hardness values of 6.35, 7.58, and 8.59 GPa for the materials HfTiZr, (HfTiZr)98Dy2, and (HfTiZr)96Dy2Nd2, respectively. The observed rise in hardness can be attributed to the concurrent increase in solid solution hardening, which arises from the introduction of larger atomic radii, leading to an increase in lattice distortion. The present study presents a novel approach to include rare earth elements into ternary refractory alloys, thereby facilitating the formation of a dual-phase structure and leading to improved mechanical properties.